Capillary waves on a falling film

Abstract

This paper shows that the wavelength and rate of spatial attenuation of the small ripples accompanying the large waves that form on a liquid film flowing down an inclined plane are dictated by the requirement that their velocity equal the velocity of the large wave. For this purpose numerical simulations of film flow are carried out and validated by comparison with published work. The wavelength and rate of spatial attenuation of the computed capillary waves are shown to equal, to an excellent approximation, the corresponding quantities obtained from an approximate form of the Orr-Sommerfeld equation in which the phase velocity is set equal to the velocity of the large waves. Thus the overall structure of the falling film waves is similar to that encountered in many other problems exhibiting a wave-hierarchical structure, such as waves in dusty or reacting gases, hydraulic jumps, undular bores, and others.